Nevanlinna--Pick norms: towards a scattered--Cantor dichotomy for spectra of commutative Banach algebras
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| Format: | Preprint |
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2025
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| author | Ohrysko, Przemysław Wojciechowski, Michał |
| author_facet | Ohrysko, Przemysław Wojciechowski, Michał |
| contents | We introduce Nevanlinna--Pick norms associated with finite families of characters in a commutative semisimple Banach algebra and study the class $NP_\infty$, where all such norms are minimal. Our main result is a topological rigidity theorem: if $A\in NP_\infty$ and $K\subsetΔ(A)$ is compact scattered, then the restriction algebra $NP(A,K)$ is isometrically $C(K)$. Consequently, if $Δ(A)$ is compact scattered, then $A\in NP_\infty$ precisely when $A$ is isometrically $C(Δ(A))$ under the Gelfand transform. This applies, in particular, to ordinal intervals and one-point compactifications of generalized Mrowka spaces. Conversely, every compact Hausdorff space containing a Cantor subset occurs as the spectrum of a commutative unital Banach algebra $A\in NP_\infty$ with $A\ne C(Δ(A))$. We also discuss uniform algebras: examples with all points peak points belong to $NP_\infty$, and $NP_\infty$ is equivalent to all Gleason parts being singletons. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_19385 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nevanlinna--Pick norms: towards a scattered--Cantor dichotomy for spectra of commutative Banach algebras Ohrysko, Przemysław Wojciechowski, Michał Functional Analysis Primary 46J05, Secondary 46J10, 54G12 We introduce Nevanlinna--Pick norms associated with finite families of characters in a commutative semisimple Banach algebra and study the class $NP_\infty$, where all such norms are minimal. Our main result is a topological rigidity theorem: if $A\in NP_\infty$ and $K\subsetΔ(A)$ is compact scattered, then the restriction algebra $NP(A,K)$ is isometrically $C(K)$. Consequently, if $Δ(A)$ is compact scattered, then $A\in NP_\infty$ precisely when $A$ is isometrically $C(Δ(A))$ under the Gelfand transform. This applies, in particular, to ordinal intervals and one-point compactifications of generalized Mrowka spaces. Conversely, every compact Hausdorff space containing a Cantor subset occurs as the spectrum of a commutative unital Banach algebra $A\in NP_\infty$ with $A\ne C(Δ(A))$. We also discuss uniform algebras: examples with all points peak points belong to $NP_\infty$, and $NP_\infty$ is equivalent to all Gleason parts being singletons. |
| title | Nevanlinna--Pick norms: towards a scattered--Cantor dichotomy for spectra of commutative Banach algebras |
| topic | Functional Analysis Primary 46J05, Secondary 46J10, 54G12 |
| url | https://arxiv.org/abs/2512.19385 |